
Time-translation invariance and uniform-time prior: As mentioned earlier, a key observation is that the master equation (REF ) is time-translation invariant. More precisely it is invariant under translations of any time variable \(t\)  related to proper time via a lapse function \({\cal N}_I\)  that is a function of \(H_I\)  only. Without any additional information, the uniform prior is the most intuitive and reasonable choice.
Let us warm up using a simple but well known example: the diffusion problem of a particle in one dimension. Given \(P(x|t)\)  as the solution to the diffusion equation, which is time-translation invariant, one may ask for the posterior probability \(P(t|x)\) , which is the distribution of lapsed time \(t\)  given that the particle is observed at location \(x\) . Without further information, the most intuitive way to calculate such quantity is simply
\(P(t|x) = \frac{P(x|t)}{ \frac{1}{T}\int \limits _0^{T} {\rm d}t^{\prime } P(x|t^{\prime })}\, .\) 
By consulting Bayes' theorem, \(P(t|x)P_x(x) =P(x|t)P_t(t)\) , the intuitive answer corresponds exactly to using a uniform prior with a cutoff in time \(T\) , which can be sent to infinity. The prior \(P_t(t)\)  in this simple example can be regarded as the distribution of the time of observation. For instance, if instead one were told that most of the observations were concentrated within a particular time window, it would be appropriate in this case to use a non-uniform \(P_t(t)\) .
The Markov process described by the master equation \( \dot{f}_I = \sum _J {\mathbb {M}}_{IJ}f_J\)  is analogous to the simple diffusion problem. Given a distribution of initial condition \(p_{\alpha }\) , the solution to the master equation,
\(P(I|t) = f_I(t) = \sum _{\alpha } \left({\rm e}^{{\mathbb {M}}t}\right)_{I\alpha } p_{\alpha }\,,\) 
is exactly the conditional probability of being in vacuum \(I\)  given time \(t\) , as explained in the previous Sections. To find the distribution of lapsed time given that the state is observed in vacuum \(I\) , \(P(t|I) =\frac{P(I |t)P_t(t) }{P(I)}\) , without additional information, the most natural choice of prior \(\rho (t) =P_t(t)\)  with maximal entropy is a uniform prior:
\(\rho (t) = {\rm constant}\,.\) 
Of course, the uniform distribution on the half real line is not normalizable, and a regularization is needed. A mathematically convenient choice is
\(\rho (t) = \epsilon {\rm e}^{-\epsilon t}\,,\) 
with a cutoff \(\sim \epsilon ^{-1}\)  in time, therefore
\(P(t|I) =\frac{P(I|t) \,{\rm e}^{-\epsilon t} }{\int \limits _0^\infty {\rm d}t^{\prime } P(I|t^{\prime })\, {\rm e}^{-\epsilon t^{\prime }}}\,.\) 
At the end of the calculation we will send \(\epsilon \rightarrow 0\)  and obtain well-defined, gauge-invariant posterior probabilities.One should stress that the nature of \(\epsilon \)  is quite different than the late-time cutoff usually introduced to regularize ratios of number counts, such as (REF ). In our approach \(\epsilon \)  is only necessary to make the improper uniform prior well-defined in the intermediate steps. Importantly, none of our results are sensitive to the choice of regulator. For instance, choosing \(\rho (t) = 1/T\)  over a large but finite interval \(0 \le t \le T\) , yields identical results in the limit \(T\rightarrow \infty \) .
A geometric way to interpret this prior is to focus on one of the metastable dS vacua in our past, for instance our parent vacuum.
Assuming it is sufficiently long-lived, the parent geometry is approximately invariant under the 10 dS isometries.
Taking a random set of world-lines with initial condition \(p_\alpha \) , the chance that a randomly chosen world-line is in vacuum \(I\)  at time \(t\)  is exactly predicted by the solution to the master equation.
The action for local operators on the world-line is of the general form
\(S = \int {\rm d}\tau \,{\cal O}(X^\mu (\tau ))\,,\) 
where the measure \({\rm d}\tau \)  is proper time along the curve. After sufficient time, the world-line approaches a geodesic comoving (\(\vec{x} = {\rm const.}\) ) in the dS flat slicing:
\({\rm d}s^2 = - {\rm d}t^2 + {\rm e}^{2Ht} {\rm d}\vec{x}^2 \,.\) 
Hence \(\tau \)  coincides with cosmic proper time \(t\) , and the resulting measure \({\rm d}\tau \simeq {\rm d} t\)  on the world-line is time-independent. This simple line of reasoning suggests that the prior density \(\rho (t)\)  be (proper) time-translation invariant. Note that the prior is also quasi-uniform in e-folding time, \(\rho (N) = \frac{\epsilon }{H} {\rm e}^{- \frac{\epsilon }{H}N}\) . In terms of conformal time, it corresponds to the Jeffreys prior, \(\rho (\eta ) \sim \eta ^{-1}\) , consistent with the dS dilation symmetry \(\eta \rightarrow \lambda \eta \) , \(\vec{x}\rightarrow \lambda \vec{x}\) .
Heuristically, a time-translation invariant prior reflects complete ignorance about the time of nucleation, akin to a temporal Copernican principle. We simply do not know when eternal inflation started along our particular past world-line. Correspondingly, this is the minimally informative prior for the time of nucleation.

Volume-weighted prior: An alternative approach is to consider a finite spatial region in the parent dS geometry. The invariant measure for observables within the region is the usual volume element:
\(\sqrt{-g}\, {\rm d}t\, {\rm d}^3x = a^3(t) \,{\rm d}t\,{\rm d}^3x\,.\) 
Averaging over \(\vec{x}\)  yields a measure \(\sim a^3(t) \,{\rm d}t\)  that grows with volume. This line of reasoning suggests that the
prior density \(\rho (t)\)  should similarly grow with volume:
\(\rho (t) \sim a^3(t)\,.\) 
This choice can also be motivated heuristically as follows. A time translation in dS corresponds to an exponential growth in volume, and therefore an exponential growth in the number of observers.
If one abides by the self-indication assumption [1]}, whereby prior probabilities are weighted by the number of observers produced, then \(\rho (t)\)  should increase exponentially
in time.The self-indication assumption can lead to absurd conclusions in other contexts, such as the “presumptuous philosopher" problem [1]}. Neal instead advocates applying what he
calls Full Non-Indexical Conditioning [3]}. We will argue in Sec.  that the late-time hypothesis is disfavored by the data, hence debating the relative merits of these assumptions in eternal inflation is perhaps moot.
For the prior distribution to be normalizable, a regulator is once again necessary. This can be achieved simply by imposing a cutoff time \(t_{\rm c}\) :
\(\rho (t) = \frac{a^3(t)}{\int \limits _0^{t_{\rm c}} {\rm d}t^{\prime }\, a^3(t^{\prime })} \qquad \text{for}~~0\le t \le t_{\rm c} \,,\) 
and \(\rho (t) = 0\)  for \(t > t_{\rm c}\) . As in the previous case, we will remove the regulator (\(t_{\rm c}\rightarrow \infty \) ) at the end of the calculation and obtain well-defined, gauge-invariant posterior probabilities.
In fact, we will see that any growing \(\rho (t)\)  that dominates after the relaxation time scale of the landscape gives the same probabilities.
Physically, since \(\rho (t)\)  multiplies the occupational probabilities \(f_j(t)\)  (see (REF )), this choice amounts to weighing probabilities by physical volume.
In standard approaches to the measure problem, volume-weighing leads to exponential sensitivity on the choice of time variable [4]}, [5]}, [6]}.
In contrast, because volume-weighing is implemented through the prior density \(\rho (t)\)  in our context, the resulting probabilities will be manifestly gauge invariant.
Since the physical volume of all bubble universes grows in proper time asymptotically as \(\sim {\rm e}^{3H_{\rm max}t}\) , where \(H_{\rm max}\)  is the highest Hubble
rate of any dS vacuum in the landscape [7]}, [8]}, [6]}, (REF ) is effectively equivalent to, in proper time,
\(\rho (t) \sim {\rm e}^{3H_{\rm max}t} \,.\) 
And because \(H_{\rm max}\)  is enormous compared to transition rates governing the evolution of the \(f_j(t)\) 's, (REF ) is well-approximated by a delta function
\(\rho (t) \simeq 2\, \delta (t-t_{\rm c}) \,;\qquad 0\le t \le t_{\rm c}\,.\) 
Thus, because the prior is so sharply peaked, it is maximally informative.
We will make this statement precise below once we derive the prior predictive probabilities in this case.
The fact that \(\rho (t)\)  is sharply peaked at the cutoff time is related to the so-called “youngness paradox" [10]} afflicting global measures based
on the proper time cutoff. No such paradox arises in our framework, as far as we can tell.

